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pgrou
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There are 5 letters in PGROU ( G2O1P3R1U1 )
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In mathematical group theory, given a prime number p, a p-group is a group in which each element has a power of p as its order. That is, for each element g of a p-group, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p. * Abelian p-groups are also called p-primary or simply primary. * A finite group is a p-group if and only if its order (the number of its elements) is a power of p. Given a finite group G, the Sylow theorems guarantee for every prime power pn that divides the order of G the existence of a subgroup of G of order pn. * The remainder of this article deals with finite p-groups. For an example of an infinite abelian p-group, see Prüfer group, and for an example of an infinite simple p-group, see Tarski monster group. |